Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2011, Article ID 854719, 4 pages
doi:10.1155/2011/854719
Research Article
Justification of the NLS Approximation for the KdV
Equation Using the Miura Transformation
Guido Schneider
IADM, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Correspondence should be addressed to
Guido Schneider, guido.schneider@mathematik.uni-stuttgart.de
Received 4 March 2011; Accepted 16 March 2011
Academic Editor: Pavel Exner
Copyright q 2011 Guido Schneider. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
It is the purpose of this paper to give a simple proof of the fact that solutions of the KdV equation
can be approximated via solutions of the NLS equation. The proof is based on an elimination of
the quadratic terms of the KdV equation via the Miura transformation.
1. Introduction
The NLS equation describes slow modulations in time and space of an oscillating and
advancing spatially localized wave packet. There exist various approximation results, cf. 1–
4 showing that the NLS equation makes correct predictions of the behavior of the original
system. Systems with quadratic nonlinearities and zero eigenvalues at the wave number
k 0 turn out to be rather difficult for the proof of such approximation results, cf. 5, 6.
The water wave problem falls into this class. Very recently, this long outstanding problem
7 has been solved 8 for the water wave problem in case of no surface tension and infinite
depth by using special properties of this problem. Another equation which falls into this
class is the KdV equation. The connection between the KdV and the NLS equation has been
investigated already for a long time, cf. 9. In 10, 11 the NLS equation has been derived
as a modulation equation for the KdV equation, and its inverse scattering scheme has been
related to the one of the KdV equation. It is the purpose of this paper to give a simple
proof of the fact that solutions of the KdV equation can be approximated via solutions of
the NLS equation. Beyond things this has been shown by numerical experiments in 12. An
analytical approximation result has been given by a rather complicated proof in 5 with a
small correction explained in 6. The much simpler proof of this fact presented here is based
on an elimination of the quadratic terms of the KdV equation via the Miura transformation.
2
Advances in Mathematical Physics
Following 13 the KdV equation
∂t u − 6u∂x u ∂3x u 0
1.1
can be transferred with the help of the Miura transformation
u v2 ∂x v
1.2
via direct substitution
2v∂t v ∂x ∂t v − 6 v2 ∂x v 2v∂x v ∂2x v 6∂x v∂2x v 2v∂3x v ∂4x v
2v ∂x ∂t v − 6v2 ∂x v ∂3x v 0
1.3
into the mKdV equation
∂t v − 6v2 ∂x v ∂3x v 0.
1.4
In order to derive the NLS equation we make an ansatz
εψv x, t εA εx − ct, ε2 t eikx−ωt c.c.
1.5
for the solutions v vx, t of 1.4, where 0 < ε 1 is a small perturbation paramater.
Equating the coefficient at εeikx−ωt to zero yields the linear dispersion relation ω −k3 .
At ε2 eikx−ωt we find the linear group velocity c −3k2 and at ε3 eikx−ωt we find that the
complex-valued amplitude A satisfies the NLS equation
∂2 A −3ik∂21 A − 6ikA|A|2 .
1.6
2. Approximation of the mKdV Equation via the NLS Equation
Our first approximation result is as follows.
Theorem 2.1. Fix s ≥ 2 and let A ∈ C0, T0 , H s3 be a solution of the NLS equation 1.6. Then
there exist ε0 > 0 and C > 0 such that for all ε ∈ 0, ε0 there are solutions of the mKdV equation
1.4 such that
sup v·, t − εψv ·, tH s ≤ Cε3/2 .
t∈0,T0 /ε2 2.1
Proof. The error function R defined by vx, t εψv x, t ε3/2 Rx, t satisfies
∂t R ∂3x R − 6ε2 ∂x ψv2 R − 6ε5/2 ∂x ψv R2 − 2ε3 ∂x R3 ε−3/2 Res εψv 0,
2.2
Advances in Mathematical Physics
with
3
Res εψv E ε4 ∂31 A − 6ε4 ∂1 A|A|2 − E3 2ε3 ik ε∂1 A3 c.c.,
2.3
where E eikx−ωt . In order to eliminate the Oε3 terms we modify the previous ansatz 1.5
by adding
ε3
6ik
3iω − 3ik3
A3 E3 c.c..
2.4
After this modification the residual Resεψv is of formal order Oε4 . When evaluated in H s
there is a loss of ε−1/2 due to the scaling properties of the L2 -norm. Hence there exist ε0 > 0
and Cres > 0 such that for all ε ∈ 0, ε0 we have
sup ε−3/2 Res εψv ·, t t∈0,T0
/ε2 Hs
≤ Cres ε2 .
2.5
By partial integration we find for s ≥ 2 and all m ∈ {0, . . . , s} that
∂t
2
2
2
2
∂m
x Rx, t dx ≤ C1 ε ψv ·, t Cs1 R·, tH s
b
C2 ε5/2 ψv ·, tCs1 R·, t3H s
2.6
b
C3 ε3 R·, t4H s C4 ε2 R·, tH s ,
with ε-independent constants Cj . Hence using a ≤ 1 a2 shows that the energy yt R·, t2H s satisfies
∂t yt C5 ε2 yt C6 ε5/2 yt3/2 C7 ε3 yt2 C8 ε2 .
2.7
Rescaling time T ε2 t and using Gronwall’s inequality immediately shows the O1
boundedness of y for all T ∈ 0, T0 , respectively all t ∈ 0, T0 /ε2 . Therefore, we are done.
3. Transfer to the KdV Equation
Applying the Miura transformation 1.2 to the approximation εψv defines an approximation
εψu ε2 ψv2 ε∂x ψv εikA εx − ct, ε2 t eikx−ωt c.c. O ε2
3.1
of the solution u of the KdV equation 1.1. Since
u − εψu H s−1
v2 ∂x v − ε2 ψv2 − ε∂x ψv H s−1
≤ v − εψv H s O ε2 O ε3/2
the approximation theorem in the original variables follows.
3.2
4
Advances in Mathematical Physics
Theorem 3.1. Fix s ≥ 1 and let A ∈ C0, T0 , H s4 be a solution of the NLS equation 1.6. Then
there exist ε0 > 0 and C > 0 such that for all ε ∈ 0, ε0 there are solutions of the KdV equation 1.1
such that
sup u·, t − εψu ·, tH s ≤ Cε3/2 .
t∈0,T0 /ε2 3.3
Acknowledgment
This paper is partially supported by the Deutsche Forschungsgemeinschaft DFG under Grant
Schn520/8-1.
References
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